29.12.12 problem 331

Internal problem ID [4931]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 12
Problem number : 331
Date solved : Sunday, March 30, 2025 at 04:16:06 AM
CAS classification : [_linear]

\begin{align*} 2 \left (-x^{2}+1\right ) y^{\prime }&=\sqrt {-x^{2}+1}+\left (1+x \right ) y \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 25

ode:=2*(-x^2+1)*diff(y(x),x) = (-x^2+1)^(1/2)+(1+x)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1}{\sqrt {x -1}}+\frac {1+x}{\sqrt {-x^{2}+1}} \]

✓ Mathematica. Time used: 0.182 (sec). Leaf size: 40

ode=2(1-x^2)D[y[x],x]==Sqrt[1-x^2]+(1+x)y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {2 \sqrt {1-x^2}+c_1 \sqrt {2-2 x}}{2 (x-1)} \]

✓ Sympy. Time used: 45.703 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(1 - x**2) + (2 - 2*x**2)*Derivative(y(x), x) - (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \sqrt {2} \left (\int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {x - 1} \left (x + 1\right )}\, dx + \int \frac {y{\left (x \right )}}{\sqrt {x - 1} \left (x + 1\right )}\, dx + \int \frac {x y{\left (x \right )}}{\sqrt {x - 1} \left (x + 1\right )}\, dx\right ) = C_{1} \]